Global 3-D imaging of mantle conductivity based on inversion of observatory C-responses—II. Data analysis and results

The global 3-D electrical conductivity distribution in the mantle (in the depth range between 400 and 1600 km) is imaged by inverting C-responses estimated on a global net of geomagnetic observatories. Very long time-series (up to 51 years; 1957-2007) of hourly means of three components of the geomagnetic field from 281 geomagnetic observatories are collected and analysed. Special attention is given to data processing in order to obtain unbiased C-responses with trustworthy estimates of experimental errors in the period range from 2.9 to 104.2 d. After careful inspection of the obtained C-responses the data from 119 observatories are chosen for the further analysis. Squared coherency is used as a main quality indicator to detect (and then to exclude from consideration) observatories with a large noise-to-signal ratio. During this analysis we found that—along with the C-responses from high-latitude observatories (geomagnetic latitudes higher than 58°)—the C-responses from all low-latitude observatories (geomagnetic latitudes below 11°) also have very low squared coherencies, and thus cannot be used for global induction studies. We found that the C-responses from the selected 119 mid-latitude observatories show a huge variability both in real and imaginary parts, and we investigated to what extent the ocean effect can explain such a scatter. By performing the systematic model calculations we conclude that: (1) the variability due to the ocean effect is substantial, especially at shorter periods, and it is seen for periods up to 40 d or so; (2) the imaginary part of the C-responses is to a larger extent influenced by the oceans; (3) two types of anomalous C-response behaviour associated with the ocean effect can be distinguished; (4) to accurately reproduce the ocean effect a lateral resolution of 1°× 1° of the conductance distribution is needed, and (5) the ocean effect alone does not explain the whole variability of the observed C-responses. We also detected that part of the variability in the real part of the C-responses is due to the auroral effect. In addition we discovered that the auroral effect in the C-responses reveals strong longitudinal variability, at least in the Northern Hemisphere. Europe appears to be the region with smallest degree of distortion compared with North America and northern Asia. We found that the imaginary part of the C-responses is weakly affected by the auroral source, thus confirming the fact that in the considered period range the electromagnetic (EM) induction from the auroral electrojet is small. Assuming weak dependence of the auroral signals on the Earth's conductivity at considered periods, and longitudinal variability of the auroral effect, we developed a scheme to correct the experimental C-responses for this effect. With these developments and findings in mind we performed a number of regularized 3-D inversions of our experimental data in order to detect robust features in the recovered 3-D conductivity images. Although differing in details, all our 3-D inversions reveal a substantial level of lateral heterogeneity in the mantle at the depths between 410 and 1600 km. Conductivity values vary laterally by more than one order of magnitude between resistive and conductive regions. The maximum lateral variations of the conductivity have been detected in the layer at depths between 670 and 900 km. By comparing our global 3-D results with the results of independent global and semi-global 3-D conductivity studies, we conclude that 3-D conductivity mantle models produced so far are preliminary as different groups obtain disparate results, thus complicating quantitative comparison with seismic tomography or/and geodynamic models. In spite of this, our 3-D EM study and most other 3-D EM studies reveal at least two robust features: reduced conductivity beneath southern Europe and northern Africa, and enhanced conductivity in northeastern Chin

Global 3-D imaging of mantle electrical conductivity based on inversion of observatory C-responses—I. An approach and its verification

We present a novel frequency-domain inverse solution to recover the 3-D electrical conductivity distribution in the mantle. The solution is based on analysis of local C-responses. It exploits an iterative gradient-type method—limited-memory quasi-Newton method—for minimizing the penalty function consisting of data misfit and regularization terms. The integral equation code is used as a forward engine to calculate responses and data misfit gradients during inversion. An adjoint approach is implemented to compute misfit gradients efficiently. Further improvements in computational load come from parallelizing the scheme with respect to frequencies, and from setting the most time-consuming part of the forward calculations—calculation of Green′s tensors—apart from the inversion loop. Convergence, performance, and accuracy of our 3-D inverse solution are demonstrated with a synthetic numerical example. A companion paper applies the strategy set forth here to real dat

Mathematical Modeling in Shell Structure Analysis Tasks

Studying shell structures while accounting for all necessary factors is a significantly nonlinear problem that requires serious mathematical tools and sufficient computing capacities. We propose an approach to solving such problems based on the following numerical methods. We propose the use of the Ritz method, the best parameter continuation method, and the Euler method under static loading. Under dynamic loading, the Kantorovich and the Rosenbrock methods are used. Software implementation was carried out using the Maple analytical software package. This paper also provides examples of simulating the deformation process in shell structures

Towards Drinfeld-Sokolov reduction for quantum groups

In this paper we study the Poisson-Lie version of the Drinfeld-Sokolov reduction defined in q-alg/9704011, q-alg/9702016. Using the bialgebra structure related to the new Drinfeld realization of affine quantum groups we describe reduction in terms of constraints. This realization of reduction admits direct quantization. As a byproduct we obtain an explicit expression for the symplectic form associated to the twisted Heisenberg double and calculate the moment map for the twisted dressing action. For some class of infinite-dimensional Poisson Lie groups we also prove an analogue of the Ginzburg-Weinstein isomorphism.Comment: 30 pages, LaTeX 2

Kolmogorov Last Discovery? (Kolmogorov and Algorithmic Statictics)

The last theme of Kolmogorov's mathematics research was algorithmic theory of information, now often called Kolmogorov complexity theory. There are only two main publications of Kolmogorov (1965 and 1968-1969) on this topic. So Kolmogorov's ideas that did not appear as proven (and published) theorems can be reconstructed only partially based on work of his students and collaborators, short abstracts of his talks and the recollections of people who were present at these talks. In this survey we try to reconstruct the development of Kolmogorov's ideas related to algorithmic statistics (resource-bounded complexity, structure function and stochastic objects).Comment: [version 2: typos and minor errors corrected

Geometrical jitter and bolometric regime in photon detection by straight superconducting nanowire

We present a direct observation of the geometrical jitter in single photon detection by a straight superconducting nanowire. Differential measurement technique was applied to the 180-{\mu}m long nanowire similar to those commonly used in the technology of superconducting nanowire single photon detectors (SNSPD). A non-gaussian geometrical jitter appears as a wide almost uniform probability distribution (histogram) of the delay time (latency) of the nanowire response to detected photon. White electrical noise of the readout electronics causes broadened, Gaussian shaped edges of the histogram. Subtracting noise contribution, we found for the geometrical jitter a standard deviation of 8.5 ps and the full width at half maximum (FWHM) of the distribution of 29 ps. FWHM corresponds to the propagation speed of the electrical signal along the nanowire of $6.2\times10^{6}$ m/s or 0.02 of the speed of light. Alternatively the propagation speed was estimated from the central frequency of the measured first order self-resonance of the nanowire. Both values agree well with each other and with previously reported values. As the intensity of the incident photon flux increases, the wide probability distribution collapses into a much narrower Gaussian distribution with a standard deviation dominated by the noise of electronics. We associate the collapse of the histogram with the transition from the discrete, single photon detection to the uniform bolometric regim